On ideal class groups of cyclotomic fields

Let m be an integer $⩾$ 3 and let F be one of the fields $Qzm+$ or $Qzm$. Denote the Galois group Gal(F/$Q$) by Δ and let p be an odd prime such that p $\nmid$ |Δ|, where |Δ| denotes the order of Δ. Let Δ denote the p-part of the ideal class group of F and (E/C)_p denote the p-part of the group E of units of F modulo the subgroup C of cyclotomic units.;For F = $Qzm$ assuming certain conditions for m and p, the equality |e_ρA = is obtained, where e_ρ is the idempotent corresponding to an irreducible higher dimensional odd character ρ of Δ into $Zp$ distinct from the Teichmüller character and is the highest power of p dividing the p -adic integer $B1,r-1$, which is defined in terms of generalized Bernoulli numbers.;The method applied is an extension of Rubin's early treatment of Kolyvagin's Euler systems.;For F = $Qzm+$, the equality |e_ρΔ| = | e_ρ(E/C)_p| is proven, where e_ρ is the idempotent corresponding to an irreducible higher dimensional character p of Δ into $Zp$. Furthermore, it is shown that e_ρ (E/C)_p is a principal $Zp$[Δ]-module.